An alternative design of the two-sided CUSUM chart for monitoring the mean when parameters are estimated

In recent literature on control charts, the exceedance probability criterion has been introduced to provide a minimum in-control performance with a specified probability. In this paper we evaluate the two-sided Phase II CUSUM charts and its in-control conditional average run length (CARL_IN) distribution with respect to the exceedance probability criterion. Traditionally, the CARL_IN distribution and its parameters has been calculated by Markov Chains and simulations. We present in this paper a generalization of the Siegmund formula to calculate the CARL_IN distribution and its parameters. This closed form formula is easy and faster to apply compared to Markov Chains. Consequently, we use it to make sample size recommendations and to adjust the charting constants via the exceedance probability criterion. The adjustments are done without bootstrapping. Results show that, in order to prevent low CARL_IN values, more Phase I data are required than has been recommended in the literature. Tables of the adjusted charting constants are provided to facilitate chart implementation. The adjusted constants significantly improve the in-control performance, at the marginal cost of a lower out-of-control performance. Balancing the trade-off between the in-control and out-of-control performance is illustrated with real data and tables of charting constants.